Question

Each matrix is nonsingular. Find the inverse of each matrix. $$ \left[\begin{array}{rr} -4 & 1 \\ 6 & -2 \end{array}\right] $$

Short answer

\[ A^{-1} = \left[ \begin{array}{cc} -1 & -\frac{1}{2} \ -3 & -2 \end{array} \right] \]

Step-by-step solution

Recall the Inverse Formula for a 2x2 Matrix

The inverse of a 2x2 matrix \[ A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\] is given by the formula \[ A^{-1} = \frac{1}{ad - bc} \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right] \] provided that the determinant \( ad - bc \) is not zero.

Compute the Determinant of the Matrix

Compute the determinant of the matrix \[ A = \left[ \begin{array}{cc} -4 & 1 \ 6 & -2 \end{array} \right] \] using the formula \[ \text{det}(A) = ad - bc \]. Here, \[ a = -4, \ b = 1, \ c = 6, \ d = -2 \]. So, \[ ad - bc = (-4)(-2) - (1)(6) = 8 - 6 = 2 \].

Apply the Inverse Formula

Using the determinant calculated in Step 2, apply the inverse formula:\[ A^{-1} = \frac{1}{2} \left[ \begin{array}{cc} -2 & -1 \ -6 & -4 \end{array} \right] = \left[ \begin{array}{cc} -1 & -\frac{1}{2} \ -3 & -2 \end{array} \right] \].

Key concepts

2x2 Matrix

Matrices are a valuable tool in various fields of mathematics and engineering. A 2x2 matrix is one of the simplest forms of matrices. It contains two rows and two columns. Every element in the matrix is a number, and specific operations can be performed on these matrices.
For example, the given matrix is:
\(\begin{bmatrix}-4 & 1 \ 6 & -2 \end{bmatrix} \).
Each number has a specific place, with numbers in the first row and first column denoted by \(a_{11}\), numbers in the first row and second column denoted by \(a_{12}\), and so forth. So, when dealing with the provided matrix, the first row, first column element is -4, the first row, second column element is 1, the second row, first column element is 6, and the second row, second column element is -2.
Understanding the position and manipulation of these elements is crucial to finding the inverse or determinant of the matrix.

Determinant

The determinant of a matrix is a special number that helps to determine whether the matrix has an inverse. For a 2x2 matrix, it is calculated using the formula \(ad - bc\), where \(a\), \(b\), \(c\), and \(d\) are elements of the matrix.
For example, in the matrix \(\begin{bmatrix} -4 & 1 \ 6 & -2 \end{bmatrix}\):

• \(a = -4\)
• \(b = 1\)
• \(c = 6\)
• \(d = -2\)

Plugging these values into the formula gives us: \( (-4)(-2) - (1)(6) = 8 - 6 = 2 \).
The determinant in this case is 2.
Since the determinant is not zero, the matrix is invertible.

Inverse Formula

Once we have determined that the matrix is invertible, the next step is to find the inverse. The inverse of a 2x2 matrix is found using a specific formula: \(\begin{bmatrix} a & b \ c & d \end{bmatrix}^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix} \).
Applying this to our matrix \(\begin{bmatrix}-4 & 1 \ 6 & -2 \end{bmatrix}\), we substitute the values:

• \(a = -4\)
• \(b = 1\)
• \(c = 6\)
• \(d = -2\)

Using the previously calculated determinant 2, we have:
\(\frac{1}{2}\begin{bmatrix}-2 & -1 \ -6 & -4 \end{bmatrix} \).
Simplifying each term inside the matrix, we finally get: \(\begin{bmatrix}-1 & -\frac{1}{2} \ -3 & -2 \end{bmatrix} \).
This is the inverse of the given 2x2 matrix.